A339100 a(n) = GCD({(2*n-k)*T(n,k)+(k+1)*T(n,k+1), k=0..n}), where T(n,k) stands for A214406 (the second-order Eulerian numbers of type B).

1, 6, 1, 60, 1, 42, 1, 120, 1, 66, 1, 5460, 1, 6, 1, 4080, 1, 798, 1, 3300, 1, 138, 1, 10920, 1, 6, 1, 1740, 1, 14322, 1, 8160, 1, 6, 1, 3838380, 1, 6, 1, 270600, 1, 12642, 1, 1380, 1, 282, 1, 371280, 1, 66, 1, 3180, 1, 798, 1, 3480, 1, 354, 1, 567867300

Offset: 1

Author

Luc Rousseau, Nov 23 2020

Keywords

• nonn

Comments

Define recursively the rational fractions R_n by: R_0(x)=1; R_{n+1}(x) = (R_n(x)*x/(1-x^2))'. 2*a(n) is the maximal integer that can be factored out of the numerator of R'_n -- staying with polynomials with integer coefficients.

Empirical observations: the prime factorizations of the a(n) follow a pattern: the 2-adic valuation of a(n) is the 2-adic valuation of n; the 3-adic valuation of a(n) is (n mod 2); for p a prime >= 5, the p-adic valuation of a(n) is 0 (if p-1 does not divide n), 1 (if p-1 divides n but p does not) or 2 (if both p-1 and p divide n). So, a(n) = 1 when n is odd, and the prime factorizations of a(n) for the first few even n are:

\ p|

\ | 2 3 5 7 11 13 17 19 23 29 31 37 41 43

n \|

---+-------------------------------------------

2 | 1 1 . . . . . . . . . . . .

4 | 2 1 1 . . . . . . . . . . .

6 | 1 1 . 1 . . . . . . . . . .

8 | 3 1 1 . . . . . . . . . . .

10 | 1 1 . . 1 . . . . . . . . .

12 | 2 1 1 1 . 1 . . . . . . . .

14 | 1 1 . . . . . . . . . . . .

16 | 4 1 1 . . . 1 . . . . . . .

18 | 1 1 . 1 . . . 1 . . . . . .

20 | 2 1 2 . 1 . . . . . . . . .

22 | 1 1 . . . . . . 1 . . . . .

24 | 3 1 1 1 . 1 . . . . . . . .

26 | 1 1 . . . . . . . . . . . .

28 | 2 1 1 . . . . . . 1 . . . .

30 | 1 1 . 1 1 . . . . . 1 . . .

32 | 5 1 1 . . . 1 . . . . . . .

34 | 1 1 . . . . . . . . . . . .

36 | 2 1 1 1 . 1 . 1 . . . 1 . .

38 | 1 1 . . . . . . . . . . . .

40 | 3 1 2 . 1 . . . . . . . 1 .

42 | 1 1 . 2 . . . . . . . . . 1

Examples

			In A214406, row number 4 is:
(k=0) (k=1) (k=2) (k=3) (k=4)
  1    112   718   744   105
Now,
(2*4-0)*  1 + (0+1)*112 =  120
(2*4-1)*112 + (1+1)*718 = 2220
(2*4-2)*718 + (2+1)*744 = 6540
(2*4-3)*744 + (3+1)*105 = 4140
(2*4-4)*105 + (4+1)*  0 =  420
The GCD of {120, 2220, 6540, 4140, 420} is 60, so a(4)=60.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..1201

Crossrefs

Cf. A214406, A165886.

Programs

Mathematica

T[n_,k_]:=T[n,k]=If[n==0&&k==0,1,If[n==0||k<0||k>n,0,(4*n-2*k-1)*T[n-1,k-1]+(2*k+1)*T[n-1,k]]]
A[n_]:=Table[(2*n-k)*T[n,k]+(k+1)*T[n,k+1],{k,0,n}]/.{List->GCD}
Table[A[n],{n,1,100}]

PARI

r(n)=if(n==0,1,(r(n-1)*x/(1-x^2))')
a(n)=my(p=(r(n))'*(1-x^2)^(2*n+1)/2);p/factorback(factor(p))
for(n=1,60,print1(a(n),", "))